Two explanations can fit the same range residual
A receiver solution changes its estimated height when the atmospheric model changes. That does not automatically mean the ground moved or the atmosphere was measured incorrectly. Several parameters can explain overlapping parts of the same observations. Understanding their different elevation signatures shows when the data can separate them and when a solution depends strongly on constraints.
This guide develops that question with a small, original linear model. It begins with a known clock and two unknowns, then restores the clock as a third unknown. The worked cases show why more measurements at one elevation are not the same as more independent geometric information.
The numerical delays, heights, angles and uncertainties are fictional teaching values. The simple mapping function is an educational approximation, not an operational atmosphere model. Use the linked specialist article for converting GNSS wet delay into water-vapour quantities; here the focus is whether positioning observations can distinguish the parameters in the first place.
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Choose your reading route
Route 1: Understand the competing parameters
Separate the radio observation from the estimated atmosphere
Route 2: Solve height and zenith delay
Route 3: Add clocks and uncertainty
Route 4: Investigate modelling choices
Balance lower-elevation information against model sensitivity
Route 5: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Understand the competing parameters
Separate the radio observation from the estimated atmosphere
Define the elevation signatures and the residual sign
Solve height and zenith delay
Solve a two-elevation case with a known clock
Make the trade-off visible with a deliberately wrong constraint
See why repeated elevation does not create a new direction
Add clocks and uncertainty
Restore the receiver clock and recount the unknowns
Add a third independent elevation and solve all three
Propagate observation noise and retain parameter correlation
CHAPTER 1 OF 12
1. Separate the radio observation from the estimated atmosphere
GNSS processing interprets radio observations using geometric, timing and propagation models. A zenith delay estimate is a parameter inferred through that model, not a direct reading from a humidity sensor. Receiver coordinates, clock terms and other contributions can participate in the same estimation problem. Their separation depends on the observation geometry and the assumptions imposed.
ESA Navipedia describes neutral-atmosphere delay using hydrostatic and wet components, with separate mapping functions in precise work. The neutral-atmosphere contribution is approximately nondispersive at GNSS frequencies, so an ionosphere-free frequency combination does not remove it in the way it targets the leading ionospheric term. Frequency diversity and elevation diversity answer different questions.
For this guide, assume a baseline atmosphere model has already been applied and study a residual zenith wet-delay correction z. Also assume horizontal coordinates and other effects are known in the initial classroom model. These are deliberate simplifications that let height and wet delay be compared directly. They are not claims that real observations contain only two errors.
Before interpreting any fitted value, ask whether it is a total zenith delay, a wet component or a residual adjustment to a prior model. A numerical result labelled atmosphere is incomplete. Its meaning depends on what was already removed and which remaining parameters were estimated alongside it. Correct units do not repair an undefined parameter.
CHAPTER 2 OF 12
2. Define the elevation signatures and the residual sign
Let E be satellite elevation above the local horizon. Define r as observed minus predicted range after the baseline model. Let h be a small upward correction to receiver height, z a residual zenith wet-delay correction and b a receiver-clock range correction. Our simplified linear equation is r(E) = −sin(E)h + m(E)z + b.
The height coefficient is negative because an upward receiver displacement shortens the geometric path towards an above-horizon satellite to first order, with vertical projection sin(E). The atmospheric coefficient m(E) maps a zenith correction into a slant-path correction. A positive clock range term adds equally to every observation in this simplified same-epoch model.
For hand calculations only, choose m(E) = 1/sin(E), the plane-parallel slab approximation. At the zenith, E = 90 degrees and m = 1. At E = 30 degrees, sinE = 0.5 and m = 2. Lower elevation lengthens the path through this idealised horizontal slab. Near the horizon, its divergence is one reason not to treat it as an operational mapping function.
The three coefficients have different shapes with elevation: −sinE, 1/sinE and a constant one. Their differences can help separate parameters when suitable independent observations exist. But different formulas alone do not guarantee a well-conditioned solution over every observed elevation range. The actual rows of the design matrix determine how much independent information was collected.
CHAPTER 3 OF 12
3. Solve a two-elevation case with a known clock
First stipulate b = 0. Use fictional residuals r₉₀ = 0.010 metres and r₃₀ = 0.050 metres. The two equations are −h + z = 0.010 and −0.5h + 2z = 0.050. Both h and z are in metres, and the coefficients are dimensionless.
From the first equation, z = h + 0.010. Substitute into the second: −0.5h + 2h + 0.020 = 0.050, so 1.5h = 0.030 and h = 0.020 metres. Then z = 0.030 metres. The solution contains a 20-millimetre height correction and a 30-millimetre zenith wet-delay correction within the stated model.
Check by reconstruction. At 90 degrees, −20 + 30 = 10 millimetres. At 30 degrees, −0.5 × 20 + 2 × 30 = −10 + 60 = 50 millimetres. The same two parameters reproduce both observations. The arithmetic demonstrates identifiability for this idealised two-unknown geometry, not evidence that a real receiver moved 20 millimetres.
One residual alone could not determine both quantities. At the zenith, h = 0 and z = 10 millimetres fits the first observation, as does h = 20 and z = 30. The second elevation rules out many of those alternatives because height and wet delay scale differently there. The information comes from the contrasting coefficients, not simply from having two rows in a table.
CHAPTER 4 OF 12
4. Make the trade-off visible with a deliberately wrong constraint
Return to the same fictional observations but force z = 0, as though the baseline atmosphere were known perfectly. The zenith residual then implies h = −0.010 metres. The 30-degree residual implies h = −0.100 metres. A single height correction cannot satisfy both equations under the forced atmospheric assumption.
If an equal-weight least-squares calculation nevertheless fits only h, its estimate is the sum of aᵢrᵢ divided by the sum of aᵢ², where aᵢ = −sinEᵢ. The numerator is −1 × 0.010 − 0.5 × 0.050 = −0.035, and the denominator is 1 + 0.25 = 1.25. The fitted h is −0.028 metres.
This constrained result is not the original +0.020-metre height correction. The omitted atmospheric degree of freedom has distorted the fitted parameter. Its reconstructed residuals are 0.028 at the zenith and 0.014 at 30 degrees, leaving observed-minus-fitted discrepancies of −0.018 and +0.036 metres. A completed computation has not made the restricted model adequate.
Now force h = 0 instead. Each observation implies a different zenith correction: 0.010 at the zenith and 0.025 at 30 degrees. The lesson is symmetrical: a parameter constraint can transfer unexplained structure into another parameter or leave systematic residuals. Constraints can be useful, but their uncertainty and scientific justification should remain visible.
CHAPTER 5 OF 12
5. See why repeated elevation does not create a new direction
Suppose every useful satellite observation in the simplified known-clock problem has E = 30 degrees. Every row is then r = −0.5h + 2z. Ten or a thousand such observations can improve knowledge of that combination under an appropriate noise model, but they do not uniquely separate h and z. The coefficient rows point in the same parameter direction.
A family of equally fitting changes is Δh = 4Δz, because −0.5 × 4Δz + 2Δz = 0. If one solution fits, adding any member of that family leaves every predicted observation unchanged. This is an exact rank deficiency in the classroom model, not merely large random uncertainty that more identical rows will eliminate.
Use a fictional observed combination of 0.040 metres. One solution is h = 0 and z = 0.020. Another is h = 0.040 and z = 0.030. Both give −0.5h + 2z = 0.040. The observations can distinguish neither solution without another elevation, external information or an explicit constraint.
Real geometry includes azimuth, multiple coordinate components and changing conditions, so this one-dimensional height demonstration is not a complete network analysis. Its transferable result is exact: repeated measurements of one parameter combination do not automatically identify its separate components. Count independent sensitivities, not merely satellites or samples.
CHAPTER 6 OF 12
6. Restore the receiver clock and recount the unknowns
Now let b be unknown. The original two observations become −h + z + b = 0.010 and −0.5h + 2z + b = 0.050. There are three unknowns but only two independent equations. A numerical routine may return one constrained solution, but the observations alone do not uniquely identify all three.
Subtract the first equation from the second: 0.5h + z = 0.040. Choose h freely, compute z = 0.040 − 0.5h, then b = 0.010 + h − z. The resulting family can also be written b = −0.030 + 1.5h. At h = 0.020, it gives z = 0.030 and b = 0, recovering the earlier known-clock solution.
At h = 0, however, it gives z = 0.040 and b = −0.030. Check both observations: 0.040 − 0.030 = 0.010 at the zenith, and 0.080 − 0.030 = 0.050 at 30 degrees. Different height and atmosphere estimates fit exactly because the clock parameter supplies additional freedom.
This is why a statement such as two elevations separate height and wet delay must retain its known-clock assumption. Restoring an omitted unknown changes the problem. Understanding a worked example includes recognising which conclusion no longer follows when its simplifying conditions are removed. The arithmetic can remain correct while its scope becomes too broad.
CHAPTER 7 OF 12
7. Add a third independent elevation and solve all three
Choose a third fictional elevation whose sine is 0.8, so the educational mapping factor is 1.25. Keep h = 0.020 metres, z = 0.030 metres and b = 0 as the generating values. The new residual is −0.8 × 0.020 + 1.25 × 0.030 = −0.016 + 0.0375 = 0.0215 metres.
Subtract the zenith equation from this third equation: 0.2h + 0.25z = 0.0115. Together with the earlier difference equation 0.5h + z = 0.040, this supplies two independent equations for h and z. Substitute z = 0.040 − 0.5h to obtain 0.2h + 0.010 − 0.125h = 0.0115. Thus 0.075h = 0.0015 and h = 0.020.
Then z = 0.030 and b = 0. The third elevation has removed the exact ambiguity in this classroom configuration. Merely adding another observation at 30 or 90 degrees would not have added the same independent row. The chosen coefficient pattern matters, not the label third satellite.
The relatively small coefficient 0.075 also hints at sensitivity. Small changes in the difference equations can affect the recovered height noticeably. Unique does not mean precise. A full uncertainty analysis should inspect weighting, covariance and conditioning; exact solution of noiseless synthetic values is only the beginning of understanding an estimation problem.
CHAPTER 8 OF 12
8. Propagate observation noise and retain parameter correlation
Return to the known-clock two-elevation problem. Its algebra gives h = (r₃₀ − 2r₉₀)/1.5 and z = (r₃₀ − 0.5r₉₀)/1.5. Suppose the two residual errors are independent with equal standard uncertainty σ = 0.003 metres. These are stipulated uncertainties for this exercise, not actual receiver specifications.
The height variance is (1² + 2²)σ²/1.5² = 5σ²/2.25. Its standard uncertainty is √5/1.5 × 0.003 ≈ 0.004472 metres. The wet-delay variance is (1² + 0.5²)σ²/2.25 = 1.25σ²/2.25, giving standard uncertainty approximately 0.002236 metres.
Their covariance is [1 × 1 + (−2) × (−0.5)]σ²/2.25 = 2σ²/2.25 = 0.000008 square metres. Dividing by the product of the two standard uncertainties gives correlation 0.8. Height and wet-delay errors tend to move together under this observation configuration, even though the input residual errors were assumed independent.
The sign makes physical sense within our model: increasing positive h shortens the predicted range, while increasing z lengthens it. They can compensate partially. Reporting separate uncertainty numbers without the covariance loses that relationship. It can distort uncertainty for a later derived quantity that uses both parameters, even when each marginal standard uncertainty was calculated correctly.
CHAPTER 9 OF 12
9. Balance lower-elevation information against model sensitivity
In the slab illustration, lower elevations increase m(E) while decreasing the magnitude of the height coefficient sinE. This can create useful contrast between parameter signatures. It is tempting to conclude that adding the lowest possible observations must always improve the estimate. That conclusion ignores the quality and completeness of the observation model.
Low-elevation signals may be more affected by obstruction, multipath and atmospheric modelling limitations. The simple slab approximation also becomes increasingly unsuitable near the horizon. An observation can add nominal geometric leverage while introducing poorly modelled error. Whether it improves the intended result depends on its uncertainty, dependence and the actual propagation model.
Use a fictional comparison. One added observation has a useful coefficient direction but an unmodelled systematic residual of 30 millimetres. Treating it as a 3-millimetre independent observation gives it unjustified influence. Its formal leverage is not evidence that the measurement is trustworthy. Conversely, excluding all contrasting elevations can leave parameters weakly separated. The task is a justified trade-off, not a universal cutoff slogan.
A meaningful comparison examines parameter stability, residual structure and independent evidence under documented processing alternatives. Do not select the solution merely because it yields the preferred height or smallest formal uncertainty. A lower uncertainty computed under an unrealistic noise model can be less informative than a larger, defensible uncertainty with visible limitations.
CHAPTER 10 OF 12
10. Distinguish a mapping error from a water-vapour conversion error
A mapping function connects zenith delay with slant-path delay at a given viewing geometry. A water-vapour conversion connects an estimated zenith wet delay with a meteorological column quantity under additional temperature-related assumptions. These are different transformations with different inputs and failure modes.
The existing GNSS water-vapour guide develops total, hydrostatic and wet delay, pressure bias and conversion to precipitable water. Read it when the uncertain step is what a reported wet-delay value means meteorologically. This article asks an earlier question: how might range observations separate that delay estimate from coordinate and clock parameters?
An incorrectly modelled hydrostatic component can leave structure that a residual wet-delay parameter partly absorbs. That does not make the absorbed value a pure measurement of vapour. Similarly, an elevation-dependent mapping error can influence fitted height. Naming the output wet delay does not prove that every contribution entering it was caused by water vapour.
For a changed-context check, imagine two solutions using the same estimated z but different water-vapour conversion factors. Their meteorological values can differ without changing the fitted geometry. Now imagine different mapping functions producing different z and h from the same observations. That discrepancy enters earlier in the chain. Diagnose the interface where the change occurred rather than attributing every difference to humidity.
CHAPTER 11 OF 12
11. Use independent information without pretending it is exact
An external height estimate can help constrain a weak height–atmosphere trade-off, but it has its own reference, epoch, physical endpoint and uncertainty. A value copied from a map may not represent the receiver’s antenna reference point or ellipsoidal height. Introducing an incompatible external constraint can create a more stable-looking solution with a new systematic error.
In the known-clock 30-degree-only example, suppose h is independently constrained to 0.040 metres with standard uncertainty 0.010, while the measured combination r has standard uncertainty 0.004. Solving z = (r + 0.5h)/2 gives z = 0.5r + 0.25h. If the two inputs are independent, its variance is 0.25Var(r) + 0.0625Var(h).
With r = 0.040, the estimated z is 0.030 metres. Its standard uncertainty is √[0.25 × 0.004² + 0.0625 × 0.010²] ≈ 0.003202 metres. Treating the external height as exact would omit the second contribution and report only 0.002 metres. The tighter-looking number would hide uncertainty imported through the constraint.
Dependence can matter too. If the external height came from the same GNSS observations or shared processing model, independence may be unjustified. Record the evidence source and covariance account where relevant. External information is useful because it adds defensible knowledge, not merely because it appears in a different file.
CHAPTER 12 OF 12
12. Report identifiability, uncertainty and the physical conclusion separately
A complete report distinguishes three claims: the parameters are identifiable under the chosen model; their uncertainties are acceptable under the adopted observation account; and their physical interpretation is supported by independent context. None follows automatically from the preceding claim. An invertible matrix does not prove that every unmodelled effect is negligible.
For the two-elevation known-clock teaching case, the model uniquely recovers h = 20 millimetres and z = 30 millimetres. Under the separate stated independent-noise exercise, their standard uncertainties are approximately 4.47 and 2.24 millimetres with correlation 0.8. Those conditional statements should not be expanded into a real-site movement or humidity claim.
For actual work, preserve the mapping models, elevation selection, observation weights, clock treatment, antenna references and parameter constraints. Retain covariance and relevant residual diagnostics rather than only a fitted height and one atmospheric number. A later reviewer should be able to identify which information separated the parameters and which assumptions supplied the remaining support.
The transferable lesson is that two explanations can share the same observations. Good measurement seeks the geometric or external evidence that separates them, and admits when that evidence is missing. This is a richer standard than simply obtaining a numerical solution: it connects the solution with what the data can genuinely distinguish.
Independent worked case and evidence checks
Solve a new fictional known-clock case with r₉₀ = −0.004 metres and r₃₀ = 0.022 metres. Use h = (r₃₀ − 2r₉₀)/1.5 to obtain (0.022 + 0.008)/1.5 = 0.020 metres. Then z = h + r₉₀ = 0.016 metres. Reconstruct both observations: −0.020 + 0.016 = −0.004, and −0.010 + 0.032 = 0.022.
Now restore an unknown clock b. One alternative fitting the same two observations is h = 0, z = 0.026 and b = −0.030 metres. Check: 0.026 − 0.030 = −0.004 at the zenith and 0.052 − 0.030 = 0.022 at 30 degrees. The two observations alone no longer establish the original height correction.
Add an independent elevation with sinE = 0.8. The first parameter set predicts −0.8 × 0.020 + 1.25 × 0.016 = 0.004 metres. The alternative predicts 1.25 × 0.026 − 0.030 = 0.0025 metres. Their difference is only 1.5 millimetres. If the new observation uncertainty is appreciably larger, exact algebraic distinction does not imply strong practical discrimination.
Explain what additional information would be useful: a defensible clock constraint, improved independent observation geometry, a compatible external height reference or a validated atmospheric constraint. Do not claim that collecting many copies of the original two rows automatically supplies the missing third direction. The question is what new information changes the competition between explanations.
An additional covariance exercise uses the known-clock parameter uncertainties from chapter eight. Consider the derived combination q = z − 0.5h. Its variance is Var(z) + 0.25Var(h) − Cov(h,z). Substituting 0.000005, 0.000020 and 0.000008 square metres gives 0.000002 square metres, so its standard uncertainty is approximately 1.414 millimetres. Keeping only the separate uncertainties and assuming independent parameters would instead give about 3.162 millimetres.
Reverse the combination to p = z + 0.5h. The covariance contribution changes sign, so its variance becomes 0.000005 + 0.000005 + 0.000008 = 0.000018 square metres. Its standard uncertainty is approximately 4.243 millimetres. The same marginal uncertainties produce very different uncertainties for two derived combinations. Covariance is therefore information about the fitted solution, not a decorative statistical supplement.
These calculations do not establish the uncertainty of an actual atmospheric retrieval. They show how a supported covariance model affects a later use of the estimated parameters. If the observation errors were correlated, their covariance would first need to enter the parameter calculation. Carrying a matrix faithfully is valuable only when its assumptions and the quantities it describes remain clear.
A final teaching check changes the residual definition to predicted minus observed. The observation signs and corresponding parameter equation must change consistently. Retaining the old equation while merely reversing the residual column would infer opposite corrections. Sign conventions are harmless when translated completely and consequential when only half the interface is changed.
Worked comparison
| Observation arrangement | What the toy model can distinguish | Important qualification |
|---|---|---|
| One elevation, known clock | One combination of height and zenith delay | Repeating the row does not separate both |
| Two distinct elevations, known clock | Height and zenith delay in the worked configuration | Precision still depends on noise and conditioning |
| Two elevations, unknown clock | Only two combinations of three parameters | A unique solution requires additional support |
| Three independent rows, unknown clock | All three in the constructed noiseless example | Unique does not mean well determined |
| External height constraint | Adds information to the trade-off | Reference, uncertainty and dependence must be retained |
Questions and practical checks
Does an ionosphere-free combination remove neutral-atmosphere delay?
Not in the same way it cancels the leading dispersive ionospheric term. The approximately nondispersive neutral-atmosphere contribution remains part of the GNSS modelling problem.
Why are height and wet-delay errors correlated?
Their elevation signatures can partly compensate in the observation equations. The worked known-clock case produces parameter correlation 0.8 even from independent input residual errors.
Are more satellites at the same elevation useless?
No. They can improve knowledge of observed combinations and add information in a fuller model. But identical coefficient rows do not separately identify parameters that the simplified model leaves confounded.
Should all low-elevation observations be included?
There is no universal answer here. Geometric leverage must be balanced against observation quality, multipath and propagation-model limitations using appropriate processing evidence.
Can these zenith delays be read directly as rainfall?
No. Wet-delay conversion to atmospheric water requires additional assumptions, and column water is not a rainfall forecast. Use the existing water-vapour guide for that separate interpretation.
Continue with the connected subjects
Connect the atmospheric and observation meanings
Interpret zenith wet delay, pressure bias and precipitable water
Build the GPS geometry and receiver-clock model
Check what an ionosphere-free frequency combination preserves
Investigate alternative sources of apparent height change
Test multipath geometry and persistent offsets
Keep antenna endpoints and height reductions compatible
Carry shared reference observations into the covariance model
Sources and scope
ESA Navipedia: hydrostatic and wet zenith delays and their slant mapping
ESA Navipedia: precise-positioning terms and joint atmospheric modelling
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 31 October 2026.
Continue through the observation model
Follow the next guide that matches the unresolved part of the observation. These companions address different model terms; none replaces an independent accuracy check.
Orientation and phase: distinguish wind-up, ambiguity and observation continuity
Emission timing: recover the transmitting event before evaluating its state
Rotating axes: express the emission event in compatible reception-frame coordinates